Critical visibility forN-particle Greenberger-Horne-Zeilinger correlations to violate local realism

A sequence of Bell inequalities for $N$-particle systems, which involve three settings of each of the local measuring apparatuses, is derived. For Greenberger-Horne-Zellinger states, quantum mechanics violates these inequalities by factors exponentially growing with $N$. The threshold visibilities of the multiparticle sinusoidal interference fringes, for which local realistic theories are ruled out, decrease as ${\mathrm{}(2/3)\mathrm{}}^N$.